# Poisson's number

The Poisson ratio${\ displaystyle \ nu}$ (after Simeon Denis Poisson ; and Poisson's ratio , Poisson or Poisson called; symbols also ) is a material parameter in the mechanical or strength of materials . It is used to calculate the transverse contraction and is one of the elastic constants of a material . The Poisson's number is a quantity in the number dimension ; H. it is a simple number. ${\ displaystyle \ mu}$

A body that is being pulled becomes longer and thinner

If a sample (a piece of solid material of standardized size) is stretched by pulling it apart at its ends ("lengthways"), this can affect its volume. In the case of a sample whose material has a Poisson's number close to 0.5, the volume remains (almost) the same - if you pull it longer, it becomes just so much thinner that its volume remains (practically) the same (for example with rubber). A Poisson's number <0.5 means that the volume of the sample increases when it is pulled apart (all isotropic materials, for example metals). The sample becomes thinner, but not so much that the volume remains the same. A Poisson's number <0 means that the sample will get thicker as it is pulled apart.

## definition

Three-dimensional model of the transverse contraction

The Poisson's ratio is defined as the linearized negative ratio of the relative change in the dimension transverse to the uniaxial stress direction to the relative change in length when a one-dimensional mechanical stress condition is applied : ${\ displaystyle \ varepsilon _ {yy}}$${\ displaystyle \ varepsilon _ {xx}}$ ${\ displaystyle \ sigma _ {xx}}$

${\ displaystyle \ nu _ {xy} = - {\ frac {\ varepsilon _ {yy}} {\ varepsilon _ {xx}}}}$

With constant stress over the cross-section and homogeneous material properties, the following applies:

${\ displaystyle \ nu = - {\ frac {\ Delta d / d} {\ Delta l / l}}}$

where denotes the original length and the original thickness. Positive values ​​of or denote an increase in this dimension, negative values ​​correspond to a reduction. ${\ displaystyle l}$${\ displaystyle d}$
${\ displaystyle \ Delta d}$${\ displaystyle \ Delta l}$

The elastic constants are mutually related. For linear-elastic, isotropic material, the following relationship between the shear modulus , the elasticity module and the compression modulus applies : ${\ displaystyle G}$ ${\ displaystyle E}$ ${\ displaystyle K}$

${\ displaystyle \ nu = {\ frac {E} {2G}} - 1 = {\ frac {3K-E} {6K}} = {\ frac {3K-2G} {6K + 2G}}}$.

Often one also finds the formulation with the Lamé constants and : ${\ displaystyle \ lambda}$${\ displaystyle \ mu}$

${\ displaystyle \ nu = {\ frac {\ lambda} {2 (\ lambda + \ mu)}}}$.

The relative change in volume with which a body that is exclusively loaded one-dimensionally with tension (or force ) reacts to uniaxial elongation ( tensile test ) is calculated using the Poisson's ratio, neglecting quadratic terms ${\ displaystyle \ Delta V / V}$

${\ displaystyle {\ frac {\ Delta V} {V}} = (1-2 \ nu) {\ frac {\ Delta l} {l}}}$.

## Scope

The assumption of constant volume for the uniaxial stress state results in ν = 0.5. Typical values ​​of the Poisson's number are between 0.3 and 0.4 for metals and between 0.4 and 0.5 for plastics. These values ​​show that the volume and thus also the density of these materials generally change under tension / pressure.

The incompressibility is only preserved for infinitesimal deformations. In addition, there are poles in Cauchy's constitutive equations. For the calculation of almost or fully incompressible materials (e.g. rubber materials, entropy elastic materials, hyperelastic materials), Green's material models should be used.

If the Poisson's ratio is less than 0.5, the volume increases under tensile load and decreases under pressure load, because then it is ; in this case have the same sign. ${\ displaystyle 1-2 \ nu> 0}$${\ displaystyle \ Delta V {\ text {and}} \ Delta l}$

With values ​​greater than 0.5, a decrease in volume occurs under tensile load. This can be observed with various porous materials. For fiber composites or wood, Poisson's numbers greater than 0.5 also occur as a rule, since the moduli of elasticity of the three axes (x, y, z) differ. Accordingly, there are also six different Poisson's numbers, which describe the respective interaction. For example, ν xy describes the strain along the x axis due to the stress along the y axis. So that the volume of orthotropic materials would remain constant in any stress / load, all (6 in 3D) Poisson's numbers would have to be equal to 0.5, despite different moduli of elasticity.

There are also isolated linear-elastic, isotropic materials with a negative Poisson's number. Negative values ​​result in a transverse expansion instead of a transverse contraction when elongated. Such materials are called auxetic . Examples of this are certain polymer foams , crystals or carbon fibers . Taking these (rare) auxetic materials into account, the Poisson's range of values ​​for isotropic materials is expanded . ${\ displaystyle -1 <\ nu <0 {,} 5}$

## Conversion between the elastic constants

The module ... ... results from:
${\ displaystyle (K, \, E)}$ ${\ displaystyle (K, \, \ lambda)}$ ${\ displaystyle (K, \, G)}$ ${\ displaystyle (K, \, \ nu)}$ ${\ displaystyle (E, \, \ lambda)}$ ${\ displaystyle (E, \, G)}$ ${\ displaystyle (E, \, \ nu)}$ ${\ displaystyle (\ lambda, \, G)}$ ${\ displaystyle (\ lambda, \, \ nu)}$ ${\ displaystyle (G, \, \ nu)}$ ${\ displaystyle (G, \, M)}$
Compression module ${\ displaystyle K \,}$ ${\ displaystyle K}$ ${\ displaystyle K}$ ${\ displaystyle K}$ ${\ displaystyle K}$ ${\ displaystyle (E + 3 \ lambda) +}$${\ displaystyle {\ tfrac {\ sqrt {(E + 3 \ lambda) ^ {2} -4 \ lambda E}} {6}}}$ ${\ displaystyle {\ tfrac {EG} {3 (3G-E)}}}$ ${\ displaystyle {\ tfrac {E} {3 (1-2 \ nu)}}}$ ${\ displaystyle \ lambda +}$${\ displaystyle {\ tfrac {2G} {3}}}$ ${\ displaystyle {\ tfrac {\ lambda (1+ \ nu)} {3 \ nu}}}$ ${\ displaystyle {\ tfrac {2G (1+ \ nu)} {3 (1-2 \ nu)}}}$ ${\ displaystyle M-}$${\ displaystyle {\ tfrac {4G} {3}}}$
modulus of elasticity ${\ displaystyle E \,}$ ${\ displaystyle E}$ ${\ displaystyle {\ tfrac {9K (K- \ lambda)} {3K- \ lambda}}}$ ${\ displaystyle {\ tfrac {9KG} {3K + G}}}$ ${\ displaystyle 3K (1-2 \ nu) \,}$ ${\ displaystyle E}$ ${\ displaystyle E}$ ${\ displaystyle E}$ ${\ displaystyle {\ tfrac {G (3 \ lambda + 2G)} {\ lambda + G}}}$ ${\ displaystyle {\ tfrac {\ lambda (1+ \ nu) (1-2 \ nu)} {\ nu}}}$ ${\ displaystyle 2G (1+ \ nu) \,}$ ${\ displaystyle {\ tfrac {G (3M-4G)} {MG}}}$
1. Lamé constant ${\ displaystyle \ lambda \,}$ ${\ displaystyle {\ tfrac {3K (3K-E)} {9K-E}}}$ ${\ displaystyle \ lambda}$ ${\ displaystyle K-}$${\ displaystyle {\ tfrac {2G} {3}}}$ ${\ displaystyle {\ tfrac {3K \ nu} {1+ \ nu}}}$ ${\ displaystyle \ lambda}$ ${\ displaystyle {\ tfrac {G (E-2G)} {3G-E}}}$ ${\ displaystyle {\ tfrac {E \ nu} {(1+ \ nu) (1-2 \ nu)}}}$ ${\ displaystyle \ lambda}$ ${\ displaystyle \ lambda}$ ${\ displaystyle {\ tfrac {2G \ nu} {1-2 \ nu}}}$ ${\ displaystyle M-2G \,}$
Shear modulus or (2nd Lamé constant) ${\ displaystyle G}$${\ displaystyle \ mu}$
${\ displaystyle {\ tfrac {3KE} {9K-E}}}$ ${\ displaystyle {\ tfrac {3 (K- \ lambda)} {2}}}$ ${\ displaystyle G}$ ${\ displaystyle {\ tfrac {3K (1-2 \ nu)} {2 (1+ \ nu)}}}$ ${\ displaystyle (E-3 \ lambda) +}$${\ displaystyle {\ tfrac {\ sqrt {(E-3 \ lambda) ^ {2} +8 \ lambda E}} {4}}}$ ${\ displaystyle G}$ ${\ displaystyle {\ tfrac {E} {2 (1+ \ nu)}}}$ ${\ displaystyle G}$ ${\ displaystyle {\ tfrac {\ lambda (1-2 \ nu)} {2 \ nu}}}$ ${\ displaystyle G}$ ${\ displaystyle G}$
Poisson's number ${\ displaystyle \ nu \,}$ ${\ displaystyle {\ tfrac {3K-E} {6K}}}$ ${\ displaystyle {\ tfrac {\ lambda} {3K- \ lambda}}}$ ${\ displaystyle {\ tfrac {3K-2G} {2 (3K + G)}}}$ ${\ displaystyle \ nu}$ ${\ displaystyle - (E + \ lambda) +}$${\ displaystyle {\ tfrac {\ sqrt {(E + \ lambda) ^ {2} +8 \ lambda ^ {2}}} {4 \ lambda}}}$ ${\ displaystyle {\ tfrac {E} {2G}}}$${\ displaystyle -1}$ ${\ displaystyle \ nu}$ ${\ displaystyle {\ tfrac {\ lambda} {2 (\ lambda + G)}}}$ ${\ displaystyle \ nu}$ ${\ displaystyle \ nu}$ ${\ displaystyle {\ tfrac {M-2G} {2M-2G}}}$
Longitudinal module ${\ displaystyle M \,}$ ${\ displaystyle {\ tfrac {3K (3K + E)} {9K-E}}}$ ${\ displaystyle 3K-2 \ lambda \,}$ ${\ displaystyle K +}$${\ displaystyle {\ tfrac {4G} {3}}}$ ${\ displaystyle {\ tfrac {3K (1- \ nu)} {1+ \ nu}}}$ ${\ displaystyle {\ tfrac {E- \ lambda + {\ sqrt {E ^ {2} +9 \ lambda ^ {2} + 2E \ lambda}}} {2}}}$ ${\ displaystyle {\ tfrac {G (4G-E)} {3G-E}}}$ ${\ displaystyle {\ tfrac {E (1- \ nu)} {(1+ \ nu) (1-2 \ nu)}}}$ ${\ displaystyle \ lambda + 2G \,}$ ${\ displaystyle {\ tfrac {\ lambda (1- \ nu)} {\ nu}}}$ ${\ displaystyle {\ tfrac {2G (1- \ nu)} {1-2 \ nu}}}$ ${\ displaystyle M}$
Poisson's ratio for some materials ${\ displaystyle \ nu}$
material Poisson's ratio ${\ displaystyle \ nu}$
cork 0.00 (approx)
beryllium 0.032
boron 0.21
Foam 0.10 ... 0.40
Silicon carbide 0.17
concrete 0.20
sand 0.20 ... 0.45
iron 0.21 ... 0.259
Glass 0.18 ... 0.3
Si 3 N 4 0.25
steel 0.27 ... 0.30
Clay 0.30 ... 0.45
copper 0.35
aluminum 0.35
titanium 0.33
magnesium 0.35
nickel 0.31
Brass 0.37
PMMA (plexiglass) 0.40 ... 0.43
rubber 0.50
Fiber reinforced plastic
(depending on the fiber orientation)
0.05 ... 0.55
Wood (" orthotropic " material)
(depending on the fiber orientation)
0.035 ... 0.67

## Numerical values

A value of or is often assumed for metallic materials and 0.35 for thermoplastics , if no more precise values ​​are known. An error in the Poisson's ratio has significantly less effect on the calculation of the component behavior under mechanical stress than an error in the modulus of elasticity. This is why the modulus of elasticity for the material used must be precisely determined (e.g. in a tensile test), while an approximate value is often sufficient for the transverse contraction. ${\ displaystyle \ nu = 0 {,} 3}$${\ displaystyle \ nu = 1/3}$

Influences of the addition of selected glass components on the Poisson's number of a special base glass.

## The reciprocal of the Poisson's number

In geotechnics and rock mechanics , the reciprocal of the Poisson's number is also referred to as the “Poisson's number”. The symbol is then often used. A uniform designation has not yet established itself. The following rule, which Othmar Rescher proposed as early as 1965, would be recommended for standardization: In his book Static Damage Statics : Calculation and Dimensioning of Gravity Dams , he describes the Poisson's ratio with and Poisson's constant with : ${\ displaystyle m}$${\ displaystyle \ nu}$${\ displaystyle m}$

• Poisson: sign: ; with numerical values ​​from 0 to <0.5${\ displaystyle \ nu}$
• Poisson's constant or “Poisson's number” (geotechnical engineering); with numerical values> 2

where:

${\ displaystyle \ nu = {\ frac {1} {m}}}$